Problem: Simplify and expand the following expression: $ \dfrac{2}{5z - 50}+ \dfrac{5}{2z - 10}- \dfrac{5}{z^2 - 15z + 50} $
Explanation: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $5$ out of denominator in the first term: $ \dfrac{2}{5z - 50} = \dfrac{2}{5(z - 10)}$ We can factor a $2$ out of denominator in the second term: $ \dfrac{5}{2z - 10} = \dfrac{5}{2(z - 5)}$ We can factor the quadratic in the third term: $ \dfrac{5}{z^2 - 15z + 50} = \dfrac{5}{(z - 10)(z - 5)}$ Now we have: $ \dfrac{2}{5(z - 10)}+ \dfrac{5}{2(z - 5)}- \dfrac{5}{(z - 10)(z - 5)} $ The least common multiple of the denominators is: $ 10(z - 10)(z - 5)$ In order to get the first term over $10(z - 10)(z - 5)$ , multiply by $\dfrac{2(z - 5)}{2(z - 5)}$ $ \dfrac{2}{5(z - 10)} \times \dfrac{2(z - 5)}{2(z - 5)} = \dfrac{4(z - 5)}{10(z - 10)(z - 5)} $ In order to get the second term over $10(z - 10)(z - 5)$ , multiply by $\dfrac{5(z - 10)}{5(z - 10)}$ $ \dfrac{5}{2(z - 5)} \times \dfrac{5(z - 10)}{5(z - 10)} = \dfrac{25(z - 10)}{10(z - 10)(z - 5)} $ In order to get the third term over $10(z - 10)(z - 5)$ , multiply by $\dfrac{10}{10}$ $ \dfrac{5}{(z - 10)(z - 5)} \times \dfrac{10}{10} = \dfrac{50}{10(z - 10)(z - 5)} $ Now we have: $ \dfrac{4(z - 5)}{10(z - 10)(z - 5)} + \dfrac{25(z - 10)}{10(z - 10)(z - 5)} - \dfrac{50}{10(z - 10)(z - 5)} $ $ = \dfrac{ 4(z - 5) + 25(z - 10) - 50} {10(z - 10)(z - 5)} $ Expand: $ = \dfrac{4z - 20 + 25z - 250 - 50}{10z^2 - 150z + 500} $ $ = \dfrac{29z - 320}{10z^2 - 150z + 500}$